Problem 103 Solve each inequality. $$ 5-... [FREE SOLUTION]

Chapter 3: Problem 103

Solve each inequality. $$ 5-3 x \geq 9 $$

Short Answer

Expert verified

x \leq \frac{-4}{3}

Step by step solution

Subtract 5 from both sides

Start by subtracting 5 from both sides of the inequality to isolate the term involving x. \[ 5 - 3x - 5 \geq 9 - 5 \]Simplifying this, we get:\[ -3x \geq 4 \]

Divide both sides by -3

Next, divide both sides of the inequality by -3. Remember that dividing by a negative number reverses the inequality sign.\[ \frac{-3x}{-3} \leq \frac{4}{-3} \]Simplifying this, we get:\[ x \leq \frac{-4}{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebraic inequalities

Algebraic inequalities are statements that show the relationship between expressions using inequality symbols such as $ > $, $ < $, $ \geq $, and $ \leq $. Unlike equations, which state that two expressions are equal, inequalities indicate that one expression is greater than, less than, or equal to another within a range.
To solve algebraic inequalities, you need to manipulate one side of the inequality to isolate the variable, while ensuring the relationship remains valid. Understanding inequalities is crucial for many applications in mathematics and real-world scenarios, such as optimization problems in economics and engineering.

isolating variable

Isolating the variable means rearranging an equation or inequality so that the variable appears alone on one side. This process often involves several steps:

Adding or subtracting terms on both sides.
Multiplying or dividing both sides by a number.
Simplifying the resulting expressions.

For example, in the exercise, we started with the inequality: $5 - 3x \geq 9$.
To isolate $ x $, we subtract 5 from both sides: \[5 - 3x - 5 \geq 9 - 5 \] which simplifies to \[ -3x \geq 4 \]
This step-by-step method ensures that we maintain the integrity of the inequality throughout the process.

reversing inequality sign

When solving inequalities, there are special rules to remember. One crucial rule is that if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This happens because multiplying or dividing by a negative number inverts the order.
For example, from the exercise, we had: \[-3x \geq 4\]
When we divide by $ -3 $, we need to reverse the inequality sign:
\[\frac{-3x}{-3} \leq \frac{4}{-3} = x \leq \frac{-4}{3} \]
This important step ensures the inequality remains true. Keep this in mind every time you divide or multiply by a negative in an inequality as forgetting this can lead to incorrect solutions.

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91影视

Solve each inequality. $$ 5-3 x \geq 9 $$

Short Answer

Step by step solution

Subtract 5 from both sides

Divide both sides by -3

Key Concepts

algebraic inequalities

isolating variable

reversing inequality sign

One App. One Place for Learning.

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